SKEWNESS AND KURTOSIS PROPERTIES OF INCOME DISTRIBUTION MODELS. Jeff Sorensen. and

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1 roiw_ Review of Income and Wealth 011 DOI: /j x SKEWNESS AND KURTOSIS PROPERTIES OF INCOME DISTRIBUTION MODELS by James B. McDonald* Brigham Young University Jeff Sorensen University of California at Berkeley and Patrick A. Turley Harvard University This paper explores the ability of some popular income distributions to model observed skewness and kurtosis. We present the generalized beta type 1 (GB1) and type (GB) distributions skewness kurtosis spaces and clarify and expand on previously known results on other distributions skewness kurtosis spaces. Data from the Luxembourg Income Study are used to estimate sample moments and explore the ability of the generalized gamma, Dagum, Singh Maddala, beta of the first kind, beta of the second kind, GB1, and GB distributions to accommodate the skewness and kurtosis values. The GB has the flexibility to accurately describe the observed skewness and kurtosis. JEL Codes: C16, C5, E5 Keywords: skewness, kurtosis, generalized beta type distribution, generalized gamma distribution 1. Introduction Pareto s pioneering work in modeling the distribution of income was published more than a century ago. He observed that, in many cases, an approximately linear relationship existed between different income levels and the number of individuals receiving at least that level of income. While the Pareto distribution often provided an accurate model of the upper tail of the distribution, it did a poor job of describing the lower tail. Since inaccurate estimates of distributions can result in misleading policy implications, this led to the consideration of different distributions that more accurately modeled income. Gibrat s (1931) law of proportionate effect provided a theoretical foundation for the use of a two-parameter lognormal distribution, which was studied in more detail by Aitchison and Brown (1969). Battistin et al. (009) used the lognormal to compare the distribution of Note: The authors appreciate the assistance of the Luxembourg Income Study Group in providing the data used in the application in this paper. Suggestions from two anonymous referees, Richard Butler, Frank McIntyre, and David Sims are also appreciated as are comments and encouragement from Conchita D Ambrosio. *Correspondence to: James B. McDonald, 153 FOB, Brigham Young University, Provo, UT 8460, USA (James_McDonald@BYU.edu). Review of Income and Wealth 011 International Association for Research in Income and Wealth Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 DQ, UK and 350 Main St, Malden, MA, 0148, USA. 1

2 income and consumption across households. Other two-parameter models include the gamma (Salem and Mount, 1974) and the Weibull (Bartels and Van Metelen, 1975). While these two-parameter models provide increased flexibility relative to single-parameter models, they do not allow for intersecting Lorenz curves, which frequently arise with income data. Intersecting Lorenz curves can be obtained by adding a third parameter. Some common three-parameter models that have been used to model income and allow for intersecting Lorenz curves include the beta of the first kind (B1), the beta of the second kind (B), the Dagum (DAGUM), and the Singh Maddala (SM) distributions. Thurow (1970) used the B1 to explore explanatory factors associated with the distribution of income for whites and blacks in the United States. Chotikapanich et al. (011) used the B to analyze global income inequality. Dagum s (1977) distribution was based on theoretical foundations and provided a significant improved fit in many applications. Singh and Maddala s (1976) distribution also provided an improved fit relative to the two-parameter models previously considered. The generalized gamma (GG) is another three-parameter model that permits intersecting Lorenz curves and yields improved fit relative to the lognormal and gamma distributions. The generalized beta of the first and second kind (GB1 and GB, respectively) are four-parameter models that include each of the previously described models as special or limiting cases. McDonald (1984) provided an early reference to the GB1 and GB and its special cases, along with applications. Distributional characteristics, such as moments and the Gini, Pietra, and Theil measures of inequality, can be expressed in terms of the distributional parameters. Other distributions, such as the double-pareto-lognormal distributional distribution which have desirable properties and provide an excellent fit to empirical data (Kleiber and Kotz, 003; Reed and Jorgensen, 004; Reed and Wu, 008), have been recently explored; the focus of this paper will be restricted to the GB1 and GB and its special cases. There is a substantial literature describing the properties, estimation procedures, and applications of these distributions. Kleiber and Kotz s book provides an excellent summary of these issues and includes more than 500 references to the theoretical foundations and diverse applications of these and other distributions in economics and actuarial science (Kleiber and Kotz, 003). Maximum likelihood estimation is a common method of estimating the parameters of income distributions, although other methods have been used. Income data is often reported in a grouped format. Estimation with grouped data can be performed by maximizing a multinomial likelihood function or minimizing a chi-square goodness of fit statistic. Other estimators may be obtained by imposing restrictive assumptions (such as assuming that the observations appear at the midpoint of an income group) or by top coding both of which ignore intra-group variability. These restrictions can impact estimator precision. Gastwirth (197) studied the impact of grouping on estimating the Gini coefficient by deriving upper and lower bounds on the Gini coefficient. The lower bound assumes all incomes in an interval equal the average income, and the upper bound corresponds to distributing the income to maximize the spread within each group. McDonald and Ransom (1981) demonstrated that a failure to take account of sampling variation can lead to misleading results.

3 More recently, continuous income data have become increasingly available and have expanded possible estimation methods and analysis. These data include information drawn from the U.S. Census Bureau, Current Population Survey, and other sources, and they are readily available on the internet. The use of continuous observations yields more accurate estimation of such descriptive statistics as skewness, kurtosis, and Gini coefficients. In this paper we explore the ability of the GB1 and GB distributions to model skewness and kurtosis. While many of the theoretical results are available in different sources, we summarize, clarify, and expand on previously known results, and derive new skewness kurtosis spaces for the GB1 and GB distributions. In addition, we present previously unknown relationships between the skewness kurtosis spaces for different distributions. We apply the results to the Luxembourg Income Study (LIS) for 13 countries, three definitions of income, and two time periods. The GB provides the flexibility to model the observed skewness and kurtosis levels in the cases considered. The next section summarizes basic characteristics of a number of popular distributions of income (for models of positive income only). Their respective skewness kurtosis spaces are described in the Appendix. Some new results, corrections to previously published results, and known results are given. Section 3 reports the observed skewness and kurtosis values for different countries, definitions of income, and time periods and compares them to the permissible values based on the distributions considered. Section 4 summarizes our findings.. The Models Since many of the most commonly used models for the distribution of income are special cases of the generalized beta type 1 (GB1) and type (GB) distributions, we begin by defining them, their moments, and some of their special cases. The GB1 and GB probability density functions (pdfs) are defined by: GB1( y; a, b, p, q) ay 1 ( 1 ( y/ b) ) ap b B( p, q) ap a q 1, 0 < y< b ap 1 ay GB( y; a, b, p, q) b B( p, q) 1+ ( y/ b) ( ) ap a p + q, 0 < y with corresponding moments given by: E E GB GB h h b Γ( p+ q) Γ( p+ h/ a) 1 ( Y ) Γ( p+ q+ h/ a) Γ( p) h h b Γ( p+ h/ a) Γ( q h/ a) ( Y ). Γ( p) Γ( q) The Pareto distribution can be viewed as a special case of the GB1: 3

4 Paretoybp ( ;, ) GB1( ya ; 1, bpq,, 1) p 1 py, b< y p b as can the beta of the first kind (B1), used by Thurow (1970): B1( y; b, p, q) GB1( y; a 1, b, p, q) p 1 q 1 y ( b y), 0 < y< b. p bb( pq, ) The moments of the Pareto and B1 distributions can easily be obtained from expressions for the GB1 moments with appropriate substitutions. The Singh Maddala and Dagum distributions are obtained from the GB by substituting p 1 and q 1, respectively, into the GB pdf to obtain: SMyabq ( ;,, ) GB( yabp ;,, 1, q) a 1 aqy, 0 < y a a b + ( y/ b) q ( ) DAGUMyabp ( ;,, ) GB( yabpq ;,,, 1) ap 1 apy, 0 < y. ap a b + ( y/ b) p ( ) The Dagum and Singh Maddala distributions, respectively, are known as the Burr Type 3 and Burr Type 1 distributions in the statistics literature (Burr, 194; Kleiber and Kotz, 003). The beta of the second kind (B), used by Chotikapanich et al. (011), is another three-parameter special case of the GB: B( y; b, p, q) GB( y; a 1, b, p, q) p 1 y p bb( pq, )( 1+ ( yb / )) p+ q, 0 < y. The moments of the SM, Dagum, and B distributions can easily be obtained from expressions for the GB moments with appropriate substitutions. The generalized gamma (GG) was used by Kloek and van Dijk (1978), Taillie (1981), McDonald (1984), Atoda et al. (1988), and Bordley et al. (1996) to study the income distribution in a number of different countries. The GG pdf is obtained from the GB by taking the following limit: ( ) 1/ a GG ( y; a, β, p) lim q GB y; a, b q β, p, q a ap 1 ( y/ β ) ay e. ap β Γ( p) 4

5 The moments of the GG can be expressed as: E GG h h β Γ( p+ h/ a) ( Y ). Γ( p) The gamma (GAM), Weibull (W), lognormal (LN), and power function (PF) pdfs are the following special or limiting cases of the generalized gamma: GAM ( y; β, p) GG ( y; a 1, β, p) p 1 ( y/ β ) y e p β Γ( p) W( y; a, β) GG( y; a, β, p 1) ay a a ( y/ β ) 1 e a β / a aμ + 1 LN ( y; μσ, ) lim a 0 GG y; a, β( aσ), p a σ ( ln( y) μ ) σ e πyσ ( ) PF ( y; βθ, ) lim a GG ( y; β, p θ/ a) θ 1 θ y, 0 < y < β. θ β The moments of the gamma and Weibull distributions can easily be obtained from expressions for the GG moments with appropriate substitutions. The moments for the LN and PF are given by: E ( Y ) e E LN PF h hμ+ h σ / h h ( Y ) β ( ) θ. θ + h Equations for the Pietra, Theil, and Gini measures of inequality expressed in terms of the distributional parameters have been derived by various authors and are summarized in Johnson et al. (1994), Kleiber and Kotz (003), and McDonald and Ransom (008) for the LN, GG, GB1, GB, and special cases. The purpose of this paper is to consider the ability of these distributions to model observed combinations of skewness and kurtosis arising in different income studies. We use the standardized skewness and kurtosis measures respectively defined by: 5

6 γ ( ) ( ) EY ( μ ) EY 3EY μ μ γ1 3 3 σ σ ( ) ( ) + ( ) EY ( μ ) EY 4EY μ 6EY μ 3μ 4 4 σ σ where m and s in these equations denote the mean and variance of the random variable of interest. Standardized skewness and kurtosis are often denoted by ( β 1, b ) in the literature, but the notation (g 1, g ) more clearly allows for positive and negative skewness. Skewness and kurtosis can also be expressed in terms of the distributional parameters. For some pdfs the permissible skewness kurtosis combinations yield relatively simple expressions of the distributional parameters. For example, parametric expressions for feasible skewness and kurtosis combinations for the gamma can be written in terms of the distributional parameter p as γ 1 / p and g 3 + 6/p, which can be rewritten as γ 3+( 3/ ) γ1. For other distributions, tractable expressions for permissible kurtosis in terms of skewness have not been obtained, but parametric expressions for skewness and kurtosis are available. The Pareto and lognormal are two examples of distributions with fairly simple parametric representations (see the Appendix) that trace out feasible skewness kurtosis combinations in the (g 1, g ) plane. Similarly, the Weibull corresponds to a line in the (g 1, g ) plane. For the three-parameter distributions, the feasible skewness kurtosis combinations correspond to two-dimensional regions (also referred to as spaces) in the (g 1, g ) plane, which are defined by upper (U) and lower (L) bounds. Rodriguez (1977) explores feasible skewness kurtosis combinations for the SM distribution. Tadikamalla (1980) derives the upper and lower bounds defining the Dagum skewness kurtosis space and demonstrates that it includes the SM space as a proper subset. Vargo et al. (010) summarize the skewness kurtosis space corresponding to a number of distributions (including the LN, B1, B, GG, GAM, and SM) and provide expressions for bounding curves for some of the distributions. Pearson (1916) demonstrates that g (g 1) + 1 for all distributions. This inequality gives the lower bound for any empirical or theoretical distribution, and we refer to it as AD L. Klaassen et al. (000) show that g (g 1) / 15 defines a lower bound for unimodal distributions. In the Appendix, we give upper and lower skewness kurtosis bounds for the GB1 and GB, summarize and expand on previously reported results, and provide explicit expressions for the bounding curves. Figure 1 provides a graphical representation of the GB1, GB, B1, B, gamma, Pareto, lognormal, and normal skewness kurtosis spaces. The normal corresponds to the point (0,3) in the (g 1, g ) plane. The B1 and GB1 share the same skewness kurtosis lower bound (represented in the figure as B1 L and GB1 L, with bounds for other distributions labeled similarly); the lower bound for all distributions is AD L. The gamma curve provides the upper bound for the B1 skewness kurtosis space and the lower bound for the B space. The B allows for only positive skewness values, with the lower and upper bounds originating at (0,3). The 6

7 15 GB1 U B1 U GB U GB1 Pareto U LN B B U L B1 U GAM GB L 10 Kurtosis AD L GB1 L B1 L 5 GB1 U Normal Skewness Figure 1. Skewness Kurtosis Spaces for GB1, GB, B1, B, Gamma, Pareto, Lognormal, and Normal Note: The L subscript represents the lower bound and the U subscript represents the upper bound for the respective distribution s skewness kurtosis space. GB skewness values are always greater than -. For skewness values greater than -, the GB lower bound is above the GB1 lower bound; however, the GB upper bound lies above the GB1 upper bound. While not obvious from the figure, the Pareto curve is contained in the B and GB spaces, but it lies above the upper bound for the more general GB1 distribution for skewness values exceeding 3.5. This is possible since the Pareto is a special case of the GB1 with a -1, whereas the GB1, GB, and their special cases correspond to a 0 The Pareto also has a vertical asymptote at skewness of As many studies of income distributions employ special cases of the GB distribution, it is instructive to focus on the skewness kurtosis spaces for the GB and its special cases, which are depicted in Figure. The GB upper bound lies above all of the upper bounds of its special and limiting cases. The Dagum and Singh Maddala distributions share the same upper bound for positive skewness but differ slightly for negative skewness; the Dagum lower bound, however, lies below the Singh Maddala lower bound (given by the Weibull skewness kurtosis curve). The Dagum skewness kurtosis space includes the SM space as a proper subset, which helps explain why the Dagum distribution often provides a better fit than the Singh Maddala distribution. The upper bound for the GG corresponds to the LN curve for positive skewness. The GB, Dagum/SM, and B upper 7

8 15 D U SM U GG U GB U B U LNSM L W B L GAM 10 GB L D L GG L Kurtosis 5 D U Skewness Figure. Skewness Kurtosis Spaces for GB, Dagum, Singh Maddala, B, Generalized Gamma, Lognormal, Weibull, and Gamma Note: The L subscript represents the lower bound and the U subscript represents the upper bound for the respective distribution s skewness kurtosis space. bounds have vertical asymptotes at skewness values of.30, 4.8, and 5.66, respectively. The generalized gamma, Dagum, and GB all share the same lower bound, which lies above the all distribution lower bound. Not surprisingly, the GB skewness kurtosis space includes the spaces for all of its limiting and special cases. Table 1 reports upper and lower bounds (which define feasible skewness kurtosis combinations) for the B1, B, GG, Dagum, Singh Maddala, GB1, and GB. These bounds were used to construct Figures 1 and and can assist researchers in selecting an appropriate distribution. Some of the distributions, such as the B and GG, have bounds that involve the skewness and kurtosis equations for other pdfs, such as the power function (PF), inverse gamma (IGAM), and log gamma (LGAM). Other distributions, such as the Dagum and SM, have bounds that are limiting or special cases of their own skewness and kurtosis equations. It is also worth noting that many bounds are segmented into two sections one for the positive skewness part of the skewness kurtosis plane and one for the negative skewness part. The Appendix includes skewness and kurtosis equations for all the distributions considered, a summary of the definitions and related properties of the additional pdfs mentioned above, and equations for the different skewness kurtosis bounds. 8

9 TABLE 1 Bounds for Skewness Kurtosis Spaces pdf Lower Bound Upper Bound B1 (g 1, g ) AD (g 1, g ) GAM and its mirror image B (g 1, g ) GAM (g 1, g ) IGAM GG (g 1, g ) PF Negative skewness: (g 1, g ) LGAM Positive skewness: (g 1, g ) LN Dagum (g 1, g ) PF Negative skewness: lim a (g 1, g ) DAGUM Positive skewness: (g 1, g ) DAGUM with p 1 Singh Maddala (g 1, g ) Weibull Negative skewness: lim a (g 1, g ) SM Positive skewness: (g 1, g ) SM with q 1 GB1 (g 1, g ) AD Negative skewness: Mirror image of (g 1, g ) GAM to the point (-,9) and then (g 1, g ) LGAM from (-,9) to (0,3) Positive skewness: (g 1, g ) LN GB (g 1, g ) PF Negative skewness: lim p kq,a (g 1, g ) GB Positive skewness: See the Appendix Note: Equations for and details about the various bounds are found in the Appendix. To illustrate the interpretation of the results in Table 1, consider the B1 distribution. The possible combinations of (g 1, g ) that can be modeled by the B1 are defined by the region bounded below by the all distribution lower bound (γ γ1 + 1) and bounded above by the gamma skewness kurtosis curve and its mirror image ( γ 3+( 3/ ) γ1 for g 1 real). We now consider an application of these models to actual income data and investigate which pdfs are sufficiently flexible to accommodate observed skewness and kurtosis values. 3. Empirical Application: Luxembourg Income Study Data Household income data were obtained from the Luxembourg Income Study (LIS) database for 13 countries. 1 The income measures we considered were earnings, gross income, and disposable income. Two time periods were used: Wave 5 of the survey (occurring in approximately 000) and Wave 6 (occurring in approximately 004). We looked at each country having data for each of the three income definitions for the time periods considered: Australia, Canada, Denmark, Finland, Germany, Israel, Norway, Poland, Sweden, Switzerland, Taiwan, the United Kingdom, and the United States. An advantage of using the LIS dataset is that the data from each country are uniformly formatted, especially with respect to the definition of income, thus facilitating inter-country comparisons. In all cases, income was measured in nominal local currency units. Because of government regulations and privacy laws, data on individual observations cannot be downloaded. Instead, we accessed the LIS microdata through their server to calculate the sample size, mean, variance, skewness, kurtosis, and Gini coefficient for each country, income definition, and year. Another advantage of using LIS data is that the income variables are continuous, not grouped, which makes the calculation of these measures more accurate. 1 Luxembourg Income Study (LIS) Database, (multiple countries. 9

10 TABLE Definitions of LIS Income Measures Used Earnings (income before taxes and transfer payments) Gross cash wage and salary income Farm self-employment income Non-farm self-employment income Gross income (income before taxes and after transfer payments) Earnings Cash property income (includes cash interest, dividends, rents, annuities, royalties, etc.) Private occupational and other pensions Public sector occupational pensions Sickness benefits Occupational injury and disease benefits Disability benefits Maternity and other family leave benefits Military/veterans/war benefits Other social insurance benefits State old-age and survivors benefits Child/family benefits Unemployment compensation benefits Social assistance cash benefits Near-cash benefits Alimony/child support Regular private transfers Other cash income Disposable income (income after taxes and transfer payments) Gross income Minus: Mandatory contributions for self-employed (includes social security, unemployment, etc.) Mandatory employee contributions Income taxes Source: Table summarizes the definitions of income used in this study. Earnings measures income before taxes and transfer payments. Gross income measures income and transfer payments before taxes are withheld. Disposable income measures income after adjusting for taxes and transfer payments. We followed the recommendation of the LIS group and used the weighted data, which can correct for non-sampling errors and sample bias. For additional details on the weighting procedures, see Table 3 reports the sample size and distributional characteristics of interest for households reporting positive income. Not surprisingly, the income data exhibit positive skewness. There is considerable variation in the estimated values for standardized skewness and kurtosis. One questionable observation in the Sweden 005 data had a value for interest and dividends that was nearly 00 times as large as the next reported value, which greatly affected skewness and kurtosis; hence, the observation was dropped for all our analyses (see notes to Table 3). As measured by the Gini coefficient, transfer payments and taxes result in a more egalitarian distribution in 10 of the 13 countries considered, with only Australia, Taiwan, and the U.K. having similar Gini coefficients for earnings and disposable income. In all cases, taxes applied to gross income resulted in smaller Gini coefficients; however, the decrease in Switzerland and Poland was quite small. 10

11 TABLE 3 Moments and Gini Coefficients for LIS Annual Household Income Data Country Year Definition n Mean Std Dev Skew Kur Gini Australia 001 Earnings 4,510 59,838 44, Gross income 6,699 51,88 45, Disposable income 6,697 41,786 31, Earnings 6,741 64,936 49, Gross income 10,087 55,916 49, Disposable income 10,086 44,870 33, Canada 000 Earnings,98 56,95 57, Gross income 8,936 56,859 56, Disposable income 8,90 43,84 36, Earnings 1,594 61,690 63, Gross income 7,776 64,030 61, Disposable income 7,774 50,777 41, Denmark 000 Earnings 59, ,336 67, Gross income 81, ,090 84, Disposable income 81,904 43,88 163, Earnings 59,84 40, , Gross income 83,0 407,09 310, Disposable income 83,178 76, , Finland 000 Earnings 8, , , Gross income 10,40 199,933 38, Disposable income 10, , , Earnings 9,358 35,56 7, Gross income 11,6 39,31 45, Disposable income 11,0 9,86 31, Germany 000 Earnings 8,051 74,83 58, Gross income 10,98 71,75 60, Disposable income 10,98 5,68 39, Earnings 8,97 40,009 31, Gross income 11,90 39,008 55, Disposable income 11,88 9,39 49, Israel 001 Earnings 4,38 149,884 15, Gross income 5, , , Disposable income 5, ,356 98, Earnings 4,76 147, , Gross income 6,59 148, , Disposable income 6,55 10,04 116, Norway 000 Earnings 11, ,575 95, Gross income 1, ,639 40, Disposable income 1, ,334 31, Earnings 10, ,76 337, Gross income 13, , , Disposable income 13,11 360, , Poland 1999 Earnings 4,01 19,54 3, Gross income 31,73 3,141 1, Disposable income 31,53 0,600 0, Earnings 3,534,95 4, Gross income 3,03 6,84 1, Disposable income 3,07 4,414 0, Sweden 000 Earnings 10,319 91,187 65, Gross income 14, , , Disposable income 14,470 1,536 11, Earnings 11, ,616 95, Gross income 16,54 373,877 91, Disposable income 16,51 69, ,

12 TABLE 3 (continued) Country Year Definition n Mean Std Dev Skew Kur Gini Switzerland 000 Earnings 3,015 93,973 70, Gross income 3,641 96,491 76, Disposable income 3,67 73,143 60, Earnings,596 97,945 60, Gross income 3,67 98,638 61, Disposable income 3,45 73,580 44, Taiwan 000 Earnings 1, , , Gross income 13, ,14 648, Disposable income 13, , , Earnings 11,903 83,58 576, Gross income 13, ,35 681, Disposable income 13,679 87, , U.K Earnings 15,199 8,716 9, Gross income 4,944 4,589 6, Disposable income 4,830 19,596 0, Earnings 17,05 35,719 38, Gross income 7,684 31,08 34, Disposable income 7,574 4,800 7, U.S. 000 Earnings 39,61 58,669 58, Gross income 49,304 57,698 58, Disposable income 49,94 44,785 39, Earnings 6,366 63,01 65, Gross income 75,746 61,877 64, Disposable income 75,736 49,534 46, Notes: All values are expressed in units of national currency in use at time of data collection. One questionable observation in the Sweden 005 data has reported interest and dividends (which is included in gross and disposable income but not in earnings) of 1,56,993,544, whereas the next largest value is 8,448,97 and the mean value of interest and dividends, after dropping the outlier, is 8,794. Thus, the observation has been dropped. Keeping the observation results in a mean, standard deviation, skewness, kurtosis, and Gini coefficient for gross income of 374,60, 1,094,773, 1,305, 1,831,53, and 0.357, respectively. It results in a mean, standard deviation, skewness, kurtosis, and Gini coefficient for disposable income of 69,945, 758,053, 1,333, 1,883,894, and 0.38, respectively. TABLE 4 Percentage of 78 Data Points Included in Each Skewness Kurtosis Space % of Data Included in pdf Skewness Kurtosis Space GB Dagum 98.7 B 84.6 SM GB GG B Table 4 reports the percentage of the 78 cases (13 countries, 3 income definitions, time periods) included in each of the skewness kurtosis spaces considered. While maximum likelihood estimators would not match sample and theoretical moments (as would method of moments estimators), these results shed light on the 1

13 450 GB D U U SM U B U GB1 U GG U SM L B L B1 U 300 Kurtosis GB L D L GG L GB1 L B1 L Skewness Figure 3. Skewness Kurtosis Data Points and Skewness Kurtosis Spaces relative ability of the different pdfs to accommodate the observed distributional characteristics in the data considered. Only 3 of the 78 (3.85 percent) cases fall in the B1 skewness kurtosis space, whereas the B space includes 66 of the 78 (84.6 percent) cases considered. The Dagum clearly outperforms the Singh Maddala distribution, accounting for all but one of the observations. Although the GB1 lower bound lies below the GG lower bound, the two distributions perform equally well (none of the data points fall in the GB1 s extended region). Figure 3 illustrates 6 of the 78 observed skewness and kurtosis combinations along with the skewness kurtosis spaces considered. The scale of the figure was selected to facilitate distinguishing the different bounds, with 16 of the larger skewness kurtosis data points being omitted. 4. Summary and Conclusions The ability of some popular income distributions to model distributional characteristics was investigated. The GB1 and GB skewness kurtosis spaces were evaluated, and prior results on the spaces for the Pareto, lognormal, gamma, Weibull, generalized gamma, Dagum, Singh Maddala, and beta distributions were given, expanded on, and compared. Of the models considered, the GB allowed for the highest kurtosis values for positive skewness, which appears to be 13

14 important in modeling the distribution of income. The skewness kurtosis values observed for 13 countries, three definitions of income, and two time periods were able to be modeled by the GB. Of the three-parameter models, the Dagum performed the best and nearly as well as the more general GB. References Aitchison, J. and J. A. C. Brown, The Lognormal Distribution with Special Reference to its Uses in Economics, Cambridge University Press, London, Atoda, N., T. Suruga, and T. Tachibanaki, Statistical Inference of Functional Form for Income Distribution, Economic Studies Quarterly, 39, 14 40, Bartels, C. P. A. and H. van Metelen, Alternative Probability Density Functions of Income, Research Memorandum 9, Vrije University Amsterdam, Battistin, E., R. Blundell, and A. Lewbel, Why is Consumption More Log Normal than Income? Gibrat s Law Revisited, Journal of Political Economy, 117, , 009. Bordley, R. F., J. B. McDonald, and A. Mantrala, Something New, Something Old: Parametric Models for the Size Distribution of Income, Journal of Income Distribution, 6, , Burr, I. W., Cumulative Frequency Functions, Annals of Mathematical Statistics, 13, 15 3, 194. Chotikapanich, D., W. E. Griffiths, D. S. P. Rao, and V. Valencia, Global Income Distributions and Inequality, 1993 and 000: Incorporating Country-Level Inequality Modeled with Beta Distributions, Review of Economics and Statistics, forthcoming, 011. Dagum, C., A New Model of Personal Distribution: Specification and Estimation, Economie Appliquée, 30, , Farebrother, R. W., The Cumulants of the Logarithm of a Gamma Variable, Journal of Statistical Computation and Simulations, 36, 43 6, Gastwirth, J. L., The Estimation of the Lorenz Curve and Gini Index, Review of Economics and Statistics, 54, , 197. Gibrat, R., Les Inégalités Économiqués, Librairie du Recueil Sirey, Paris, Johnson, N. L. and S. Kotz, Power Transformations of Gamma Variables, Biometrika, 59, 6 9, 197. Klaassen, C. A. J., P. J. Mokveld, and B. van Es, Squared Skewness Minus Kurtosis Bounded by 186/15 for Unimodal Distributions, Statistics and Probability Letters, 50, 131 5, 000. Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, Wiley- Interscience, New York, 1994., Continuous Univariate Distributions, Vol., Wiley-Interscience, New York, Kleiber, C. and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley and Sons, Hoboken, NJ, 003. Kloek, T. and H. K. van Dijk, Efficient Estimation of Income Distribution Parameters, Journal of Econometrics, 8, 61 74, McDonald, J. B., Some Generalized Functions for the Size Distribution of Income, Econometrica, 5, , McDonald, J. B. and M. R. Ransom, An Analysis of the Bounds for the Gini Coefficient, Journal of Econometrics, 17, , 1981., The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality, in Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, Volume 5 in Economic Studies in Equality, Social Exclusion and Well-Being, Springer, New York, 008. Pearson, K., Mathematical Contributions to the Theory of Evolution, XIX; Second Supplement to a Memoir on Skew Variation, Philos. Trans. Roy. Soc. London Ser. A, 16, 49 57, Reed, W. J. and M. Jorgensen, The Double-Pareto Lognormal Distribution A New Parametric Model for Size Distribution, Communications in Statistics-Theory and Methods, 33, , 004. Reed, W. J. and F. Wu, New Four- and Five-Parameter Models for Income Distributions, in Modeling Income Distributions and Lorenz Curves, in Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, Volume 5 in Economic Studies in Equality, Social Exclusion and Well-Being, Springer, New York, 008. Rodriguez, R. N., A Guide to the Burr Type XII Distributions, Biometrika, 64, 19 34, Salem, A. B. Z. and T. D. Mount, A Convenient Descriptive Model of Income Distribution: The Gamma Density, Econometrica, 4, ,

15 Singh, S. K. and G. S. Maddala, A Function for the Size Distribution of Incomes, Econometrica, 44, , Tadikamalla, P. R., A Look at the Burr and Related Distributions, International Statistical Review, 48, , Taillie, C., Lorenz Ordering Within the Generalized Gamma Family of Income Distributions, Statistical Distributions in Scientific Work, 6, 181 9, Thurow, L. C., Analyzing the American Income Distribution, American Economic Review, 48, 61 9, Vargo, E., R. Pasupathy, and L. M. Leemis, Moment-Ratio Diagrams for Univariate Distributions, Journal of Quality Technology, 4(3), 1 11, 010. Supporting Information Additional Supporting Information may be found in the online version of this article: Appendix: Skewness Kurtosis Spaces for Select Distributions Table A.1: λ h s Used to Calculate Skewness and Kurtosis for Different pdfs Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. 15